What should our assumptions be about regions of the Universe that have
never been in causal contact?
If we look as far away as we can in one direction and as far away as we
can in the other direction we can ask
the question, have those two points (points A and B in
Fig. 4) been able to see each other.
In the standard big bang model without inflation the answer is no. Their
past light cones are the little cones
beneath points A and B. Inserting a period of inflation
during the early universe has the effect of moving
the surface of last scattering up to the line labeled "new surface of
last scattering". Points A and B then
become points A' and B'. And the apexes of their past
light cones are at points A' and B'.
These two new light cones have a large degree of intersection. There
would have been sufficient time
for thermal equilibrium to be established between these two
points. Thus, the answer to the question: "Why
are two points in opposite sides of the sky at the same temperature?"
is, because they have been in causal
contact and have reached thermal equilibrium.

Figure 4. Inflation shifts the position of
the surface of last scattering. Here we
have modified the lower panel of
Fig. 1 to show what the
insertion of an early
period of inflation does to the past light cones of two points,
A and B, at the
surface of last scattering on opposite sides of the sky.
An opaque wall of electrons - the cosmic photosphere, also known as the
surface of last scattering - is at a scale factor
a = R / Ro 0.001 when the
Universe was
1000 times smaller
than it is now and only 380, 000 years old.
The past light cones of A and B do not overlap - they have
never seen each other - they have never been in causal contact.
And yet we observe these points to be at the same temperature. This is
the horizon problem (Sect. 4.3).
Grafting an early epoch of inflation onto the big bang model moves the
surface of last scattering upward to the line
labeled "new surface of last scattering". Points A and B
move upward to A' and B'. Their new past light cones
overlap substantially. They have been in causal contact for a long time.
Without inflation there is no overlap. With inflation there is.
That is how inflation solves the problem of identical temperatures in
`different' horizons.
The y axis shows all of time. That is, the range in conformal time [0,
62] Gyr corresponds to the cosmic time range
[0, ] (conformal time
is defined by
d = dt /
R). Consequently, there is an upper limit to the size of the
observable universe.
The isosceles triangle of events within the event horizon are the only
events in the Universe
that we will ever be able to see - probably a very small fraction of the
entire universe.
That is, the x axis may extend arbitrarily far in both directions. Like
this .

Figure 5.

Five years ago most of us thought that as we waited patiently we would
be rewarded with
a view of more and more of the Universe and eventually, we hoped to see
the full extent of the inflationary bubble -
the size of the patch that inflated to form our Universe. However,
has interrupted
these dreams of unfettered empiricism.
We now think there is an upper limit to the comoving size of the
observable universe.
In Fig. 4 we see that the observable universe
(= particle horizon) in the new
standard -CDM
model approaches 62 billion light years in radius but will never extend
further. That is as large as it gets. That is as far as we will ever be
able to see. Too bad.